What is a Set?

Transcript

0:00Introduction

Hello everyone! Okay, one quick question before we start. The five best batsmen of India. Is that a set? Most students say, of course! Five players, right? Hmm. But ask ten cricket fans, and you get ten different lists. So, set or not? And look, every branch of mathematics is written in the language of sets, so in JEE this word comes back again and again. Alright, let's begin.

0:29A set: a well-defined collection

In daily life we say a team of players, a bunch of keys, a class of students. Mathematics has one word for all of them: a set. Georg Cantor, a German mathematician, created set theory. The definition: a well-defined collection of distinct objects, called its elements, or members. The key word is well-defined. What does it mean? Yes, it means: for any object, we can clearly decide whether it belongs to the collection or not. No scope for personal opinion. In or out, crystal clear. For example, the vowels of the English alphabet: the letters a, e, i, o and u. Everyone gives the same answer, so this is a set. We write its elements inside curly brackets, okay? Now, back to our puzzle. The five best batsmen of India. Set, or not? Not a set! Best depends on the person answering; your list and your friend's list will differ, right? Look, this is the trap. The players are real, but choosing them is a matter of opinion, so not a set. See words like best or difficult? Be alert!

1:38Belongs to: the symbols ∈ and ∉

Now, how do we write that something is in a set? If small a is an element of capital A, we put this curved symbol, like a rounded e, between them. Read it: a belongs to A, okay? With a slash through it: a does not belong to A, simple. Let's try. Let A be the set one, three, five, seven, nine. Three belongs to A; four does not, nice and easy. Alright, now a tricky one: five in curly brackets, the set whose only element is five. Is it an element of A? Five is in A, so yes, right? No, no, careful! The elements of A are the numbers one, three, five, seven and nine. The set containing five is not one of them, so it does not belong to A, got it? Classic JEE trap.

2:29Standard sets of numbers

Next up, the standard sets of numbers. Natural numbers, written with this double-lined N: one, two, three, and so on. Add zero, and you get the whole numbers, the double-lined W, okay? Next, the integers, written with a double-lined Z: now the negatives come in too. Minus two, minus one, zero, one, two, in both directions. Some books write capital I for integers, same thing, okay? Then the rational numbers, the double-lined Q: all numbers p over q, where p and q are integers, and q is not zero. Read the colon on screen as, such that. Clear? In decimal form, rational numbers are the decimals that terminate, or repeat. Do remember this line; we need it in example two.

3:22Real and complex numbers

Next, the real numbers, the double-lined R: the whole number line. Some reals have decimals that never end and never repeat: the irrationals, like root two, pi and e in the picture. On screen, R minus Q: the reals with the rationals taken out. So the reals are the rationals and irrationals together, okay? One more level up: the complex numbers, the double-lined C, okay? All numbers of the form a plus i b, where a and b are real, and i squared equals minus one. So the real numbers sit inside the complex numbers. Makes sense? Now look at the picture: each set sits inside the next, like boxes inside boxes. This sideways U symbol means, sits inside: N inside W, inside Z, and so on up to C, okay? And here is a topper's trick. To classify a number, find the smallest box it fits in. It belongs to every bigger box too. That's it!

4:24Traps about zero, and one about complex numbers

Now, a few traps. Watch carefully. First: is zero a natural number? In NCERT and in JEE, natural numbers start from one, so zero is not a natural number. Some books abroad include zero, so always follow the definition given in the question, okay? But zero is an integer. It is a rational number, because zero equals zero over one, and it is a real number. And zero is neither positive nor negative, okay? Students lose easy marks on this one, so remember it. Last trap: every real number is also a complex number, because any real a equals a plus zero i. So when a question asks which sets a number belongs to, never forget C, okay?

5:11Example 1: which collections are sets

Alright, example one. Which of these are sets? One: months of the year beginning with the letter J, okay? Two: the ten most talented writers of India. Three: prime numbers less than one. Four: the difficult chapters of the JEE syllabus. Five: all solutions of x squared equals four. Pause the video, and try it first. Done? Okay, let's check. Months beginning with J: January, June, July. No doubt at all, so it is a set. Two: most talented, according to whom? It is a matter of opinion, isn't it? So, not a set. Same story as our batsmen! Three: prime numbers less than one. There is no prime less than one, nothing at all. So it is not a set, right? Wait a second! Apply the test: is it well-defined? Yes! For any number, we can clearly decide whether it is a prime less than one. So it is a set, with no elements: the empty set. Four: the difficult chapters. Difficult for you, easy for your friend. Agreed? It differs from student to student, so not a set. Five: solutions of x squared equals four. Which numbers square to four? Two, and minus two. Clear membership, so it is a set: minus two and two, in curly brackets. Absolutely perfect.

6:44Example 2: classify numbers into N, Z, Q, R and C

Now a slightly harder one. Which of N, Z, Q, R and C does each number belong to? Write all that apply. Minus three, twenty two over seven, root two, zero point three six with a bar on top, root of minus nine, and root sixteen. The bar means three six repeats forever. Pause the video and try. Hint: use the box trick. Done? Okay, let's check. Minus three: smallest box? The integers, exactly! It is negative, so not natural. So minus three is in Z, Q, R and C, all four. Next, twenty two over seven. Many students think: twenty two over seven is pi, pi is irrational, so this is irrational. Right? No, no, careful! Twenty two over seven is not equal to pi. It is an integer over a non-zero integer, so it is rational: Q, R and C, okay? Root two is irrational; we saw it in our list. So only R and C, nothing smaller. Zero point three six, repeating. Repeating decimal means rational, remember? In fact, it equals thirty six over ninety nine, that is four over eleven. So, Q, R and C, okay? Root of minus nine equals three i, because i squared is minus one. So it is only in the biggest box: C, the complex numbers. Last one, root sixteen. Looks scary, right? But it is just four. Smallest box: the natural numbers. So it is in all five sets in the question, from N to C, okay? That's it, done! See how fast the box trick is?

8:38Quick revision: well-defined sets and number sets

So, quick revision. One: a set is a well-defined collection, no opinions allowed. Two: each number set sits inside the next, from N to C, and zero is not a natural number. Three: find a number's smallest box; it is in every bigger box too. Now try the practice problems on this page. You will find them easy, I promise. All the best!